{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 3.1 线性基函数模型"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "之前我们一直在考虑概率密度估计这样的非监督问题。现在我们考虑监督问题，从简单的回归问题开始。\n",
    "\n",
    "对于回归问题，我们有一个 $D$ 维的输入 $\\sf x$，和一个或多个连续的目标输出 $\\sf t$。在第一章我们遇到了多项式回归的问题，现在我们扩展这个问题，利用一组固定的非线性基函数的线性组合，得到一个线性回归模型。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 问题设定"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "假设我们有 $N$ 个观测数据 $\\{\\mathbf x_n\\}, n=1,\\dots, N$，以及对应的目标值 $\\{t_n\\}$，我们的目标是对于预测 $\\bf x$ 对应的目标值 $t$。\n",
    "\n",
    "之前的讨论我们知道，最简单的方法是找到一个函数 $y(\\mathbf x)$，使得 $y(\\mathbf x)$ 的值能够很好的模拟 $t$。\n",
    "\n",
    "从概率的角度来说，我们希望对 $p(t|\\mathbf x)$ 进行建模，因为对于 $\\mathbf x$ 来说，$t$ 具有一定的不确定性。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 线性基函数模型"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "最简单的线性回归模型即输入数据的线性组合：\n",
    "\n",
    "$$\n",
    "y(\\mathbf{x,w}) = w_0 + w_1x_1 + \\cdots + w_Dx_D\n",
    "$$\n",
    "\n",
    "其中 $\\mathbf x = (x_1,\\dots,x_D)^\\top$，这就是简单的线性回归模型（`linear regression`）。\n",
    "\n",
    "它是参数 $w_0, w_1, \\dots, w_D$ 的一个线性组合（当然，它也是 $\\bf x$ 的线性组合，但是这限制了我们的模型）。基于这个性质，我们将这个线性回归进行扩展，变成一组关于 $\\mathbf x$ 的非线性函数的线性组合：\n",
    "\n",
    "$$\n",
    "y(\\mathbf{x,w}) = w_0 + \\sum_{j=1}^{M-1} w_j \\phi_j(\\mathbf x)\n",
    "$$\n",
    "\n",
    "这里 $\\phi_j(\\mathbf x)$ 就是所谓的基函数（`basis function`）。这里限制 $w$ 的下标最大为 $M-1$，表示我们的参数数目为 $M$\n",
    "\n",
    "$w_0$ 通常叫做偏置项（`bias`）（注意不要跟统计中的 `bias` 混淆，统计中它指的是一个估计量偏离其均值的程度），对任何输入它的作用都是一样的。\n",
    "\n",
    "通常为了方便，我们定义 $\\phi_0(\\mathbf x)=1$，这样我们得到：\n",
    "\n",
    "$$\n",
    "y(\\mathbf{x,w}) = \\sum_{j=0}^{M-1} w_j \\phi_j(\\mathbf x) = \\mathbf{w^\\top \\phi(x)}\n",
    "$$\n",
    "\n",
    "其中 $\\mathbf w = (w_0,\\dots,w_{M-1})^\\top$，$\\mathbf \\phi=(\\phi_0,\\dots,\\phi_{M-1})^\\top$。\n",
    "\n",
    "在很多应用中，我们通常会对原始输入进行一些预处理或者特征选择，这些得到的特征可以被认为是基函数 $\\{\\phi_j(\\mathbf x)\\}$。\n",
    "\n",
    "虽然我们的模型叫做线性基函数模型（`linear basis function models`），但这个模型事实上可以是非线性的；它的线性体现在参数 $\\mathbf w$ 上。\n",
    "\n",
    "在之前的多项式函数模型中，我们相当于选择了这样的基函数 $\\phi_j = x^j$。多项式拟合的一个局限在于，这个函数是一个输入变量的全局函数，因此在一个区域内数据点的改变会影响到其他所有区域。为此，我们可以考虑将输入空间划分成多个区域，然后在这些区域内使用不同的多项式拟合模型，这就是所谓 `spline` 函数的想法。\n",
    "\n",
    "常用的基函数有高斯基函数：\n",
    "\n",
    "$$\n",
    "\\phi_j(x) = \\exp\\left\\{-\\frac{x-\\mu_j}{2s^2}\\right\\}\n",
    "$$\n",
    "\n",
    "\n",
    "sigmoid 基函数：\n",
    "\n",
    "\n",
    "$$\n",
    "\\phi_j(x) = \\sigma\\left\\{\\frac{x-\\mu_j}{s}\\right\\}\n",
    "$$\n",
    "\n",
    "其中\n",
    "\n",
    "$$\n",
    "\\sigma(a)=\\frac{1}{1+\\exp(-a)}\n",
    "$$\n",
    "\n",
    "等价的我们还可以用 tanh 函数，因为 $\\tanh(a) = 2\\sigma(a)-1$，所以 sigmoid 函数的线性组合与 tanh 函数的线性组合是等效的。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
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vJyIirsbbu/7p99W0aDGJ3NyayktZWRmLFi1i/HjnObYmTJjA4sWLKS8vr7H9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\n3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T3of9QC6H61H9rxWHckA/ivwPLhgwCsFATBSkRbHU3YfcNxiOJi4IknJC+wt3aoWK+X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\nZJCRDphEIpFIJBLJIPP/JHDU29g98CUAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fad9d044a10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import scipy as sp\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "%matplotlib inline\n",
    "\n",
    "_, axes = plt.subplots(1, 3, figsize=(10, 3))\n",
    "\n",
    "tt = np.linspace(-1, 1, 200)\n",
    "\n",
    "for s in np.arange(1, 11.5, 1):\n",
    "    axes[0].plot(tt, tt ** s)\n",
    "    axes[1].plot(tt, np.exp(-(tt - (s - 6)/5.0) ** 2 / (2 * 0.04)))\n",
    "    axes[2].plot(tt, 1.0 / (1 + np.exp(-(tt - (s - 6)/5.0) / 0.1)))\n",
    "\n",
    "\n",
    "axes[0].set_title('polynomial basis')\n",
    "axes[1].set_title('gaussian basis')\n",
    "axes[2].set_title('sigmoid basis')\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "另一个常用的基函数是傅里叶基，对应于一组三角函数的展开，每个基函数代表一种特定的频率成分，如果我们在空间和频率上对其进行限制，我们可以得到小波变换 `wavelets`。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3.1.1 最大似然和最小二乘"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "最小二乘的目标是最小化：\n",
    "\n",
    "$$\n",
    "E_D(\\mathbf w) = \\frac{1}{2}\\sum_{n=1}^N \\{t_n-\\mathbf w^\\top\\mathbf \\phi(\\mathbf x_n)\\}^2\n",
    "$$\n",
    "\n",
    "在之前的多项式拟合中，我们发现，我们可以从最小平方和误差和最大似然两个角度去理解回归问题的求解。现在我们仔细解释这个模型。\n",
    "\n",
    "$$\n",
    "t = y(\\mathbf{x,w}) +\\epsilon\n",
    "$$\n",
    "\n",
    "其中 $p(\\epsilon|\\beta)=\\mathcal N(\\epsilon|0,\\beta^{-1})$，于是我们有：\n",
    "\n",
    "$$\n",
    "p(t|\\mathbf{x,w},\\beta)=\\mathcal N(t|y(\\mathbf{x,w}),\\beta^{-1})\n",
    "$$\n",
    "\n",
    "在 [1.5.5](../Chap-01-Introduction/01-05-Decision-Theory.ipynb#1.5.5-回归问题的损失函数) 节，我们知道，如果我们使用平方误差函数，那么最优解为目标值 $t$ 的条件均值。所以，在高斯分布的条件下，条件分布的均值为：\n",
    "\n",
    "$$\n",
    "\\mathbb E[t|\\mathbf x] = y(\\mathbf{x,w})\n",
    "$$\n",
    "\n",
    "回到线性基函数的问题上，假设有一组独立同分布的数据 $\\mathbf X=\\{\\mathbf x_1,\\dots,\\mathbf x_N\\}$ 及其对应目标 $\\mathsf t = (t_1,\\dots, t_N)^T $。独立同分布假设下，似然函数为：\n",
    "\n",
    "$$\n",
    "p(\\mathsf t|\\mathbf{X,w},\\beta) = \\prod_{n=1}^N \\mathcal N(t_n|\\mathbf w^\\top\\mathbf \\phi(\\mathbf x_n),\\beta^{-1})\n",
    "$$\n",
    "\n",
    "只考虑参数项，对数似然为：\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\ln p(\\mathsf t|\\mathbf{w},\\beta) & = \\sum_{n=1}^N \\ln \\mathcal N(t_n|\\mathbf w^\\top\\mathbf \\phi(\\mathbf x_n),\\beta^{-1}) \\\\\n",
    "& = \\frac{N}{2}\\ln \\beta - \\frac{N}{2}\\ln (2\\pi) - \\beta E_D(\\mathbf w)\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "其中 $E_D(\\mathbf w)$ 是平方和误差：\n",
    "\n",
    "$$\n",
    "E_D(\\mathbf w) = \\frac{1}{2}\\sum_{n=1}^N \\{t_n-\\mathbf w^\\top\\mathbf \\phi(\\mathbf x_n)\\}^2\n",
    "$$\n",
    "\n",
    "如果只考虑对 $\\bf w$进行优化，那么最大似然就相当于一个最小二乘问题。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 参数 $w$ 的最大似然"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "考虑对 $\\bf w$ 的梯度：\n",
    "\n",
    "$$\n",
    "\\triangledown \\ln p(\\mathsf t|\\mathbf{w},\\beta) = \\beta \\sum_{n=1}^N \\{t_n-\\mathbf w^\\top\\mathbf \\phi(\\mathbf x_n)\\} \\mathbf\\phi(\\mathbf x_n)^\\top = 0\n",
    "$$\n",
    "\n",
    "我们有：\n",
    "\n",
    "$$\n",
    "0=\\sum_{n=1}^N t_n \\mathbf\\phi(\\mathbf x_n)^\\top - \\mathbf w^\\top\\left( \\sum_{n=1}^N\\mathbf \\phi(\\mathbf x_n)\\mathbf\\phi(\\mathbf x_n)^\\top\\right)\n",
    "$$\n",
    "\n",
    "如果记\n",
    "\n",
    "$$\n",
    "\\mathbf\\Phi = \\begin{pmatrix}\n",
    "\\mathbf\\phi(\\mathbf x_1)^\\top \\\\\n",
    "\\mathbf\\phi(\\mathbf x_2)^\\top \\\\\n",
    "\\vdots \\\\\n",
    "\\mathbf\\phi(\\mathbf x_N)^\\top \\\\\n",
    "\\end{pmatrix}  = \\begin{pmatrix}\n",
    "\\phi_0(\\mathbf x_1) & \\phi_1(\\mathbf x_1) & \\cdots & \\phi_{M-1}(\\mathbf x_1) \\\\\n",
    "\\phi_0(\\mathbf x_2) & \\phi_1(\\mathbf x_2) & \\cdots & \\phi_{M-1}(\\mathbf x_2) \\\\\n",
    "\\vdots & \\vdots  & \\ddots & \\vdots\\\\\n",
    "\\phi_0(\\mathbf x_N) & \\phi_1(\\mathbf x_N) & \\cdots & \\phi_{M-1}(\\mathbf x_n)\\\\\n",
    "\\end{pmatrix}\n",
    "$$\n",
    "\n",
    "即 $\\mathbf\\Phi_{nj}=\\phi_j(\\mathbf x_n)$，\n",
    "则有：\n",
    "\n",
    "$$\n",
    "\\mathbf w_{ML} = {\\bf(\\Phi^\\top\\Phi)}^{-1}{\\bf \\Phi^\\top t} = \\mathbf \\Phi^\\dagger\n",
    "$$\n",
    "\n",
    "其中 $\\mathbf \\Phi^{\\dagger}={\\bf(\\Phi^\\top\\Phi)}^{-1}{\\bf \\Phi^\\top t}$ 是 $\\bf\\Phi$ 的 `Moore-Penrose` 伪逆。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "### 偏置参数 $w_0$ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "我们将偏置参数 $w_0$ 单独拿出来考虑。平方和误差可以改写为：\n",
    "\n",
    "$$\n",
    "E_D(\\mathbf w) = \\frac{1}{2}\\sum_{n=1}^N \\{t_n-w_0-\\sum_{j=1}^{M-1} w_j\\phi_j(\\mathbf x_n)\\}^2\n",
    "$$\n",
    "\n",
    "令其对 $w_0$ 的梯度为 0，我们解得：\n",
    "\n",
    "$$\n",
    "w_0 = \\bar{t}-\\sum_{j=1}^{M-1}w_j\\overline{\\phi_j}\n",
    "$$\n",
    "\n",
    "其中：\n",
    "\n",
    "$$\n",
    "\\bar{t} = \\frac{1}{N} \\sum_{n=1}^N t_n, \\overline{\\phi_j} = \\frac{1}{N} \\sum_{n=1}^N \\phi_j(\\mathbf x_n)\n",
    "$$\n",
    "\n",
    "所以偏置项是目标与基函数的加权和之间的平均误差。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "### 参数 $\\beta$ 的最大似然"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "令对数似然对 $\\beta$ 的偏导为 0：\n",
    "\n",
    "$$\n",
    "\\frac{1}{\\beta_{ML}} = \\frac{1}{2}\\sum_{n=1}^N \\{t_n-\\mathbf w^\\top\\mathbf \\phi(\\mathbf x_n)\\}^2\n",
    "$$\n",
    "\n",
    "所以它的倒数就是回归的残差平方和。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3.1.2 最小二乘的几何表示"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "考虑所有目标值组成的 $N$ 维空间，$\\mathbf t=(t_1,\\dots,t_N)^\\top$ 是这个空间的一个向量。每个基函数 $\\phi_j$，作用在 $N$ 个数据点上所形成的 $N$ 维向量 $\\mathbf\\varphi_j=(\\phi_j(\\mathbf x_1),\\dots,\\phi_j(\\mathbf x_N))^\\top$，也是这个空间的一个向量，这样的向量共有 $M$ 个。\n",
    "\n",
    "如果 $M<N$，那么这 $M$ 个向量张成的空间 $\\cal S$ 是 $N$ 维空间的一个 $M$ 维子空间。\n",
    "\n",
    "如果定义 $\\mathbf y=\\sum_{j=0}^{M-1} w_j\\varphi_j$，最小二乘项对应于 $\\mathbf y$ 和 $\\mathbf t$ 在这个空间的欧氏距离，所以 $\\mathbf y$ 是超平面 $\\cal S$ 上距离 $\\bf t$ 最近的点。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3.1.3 序贯学习"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "当数据量很大时，我们可以考虑将这个算法序列化，即所谓的在线算法。每次处理一个数据点，更新一次模型参数。\n",
    "\n",
    "为了使用在线算法，我们可以考虑使用随机梯度下降（`stochastic gradient descent`）的方法。\n",
    "\n",
    "我们更新的策略为：\n",
    "\n",
    "$$\n",
    "\\mathbf w^{(\\tau+1)} = \\mathbf w^{(\\tau)} - \\eta \\triangledown E_n \n",
    "$$\n",
    "\n",
    "$E_n$ 表示 $(\\mathbf x_n, t_n)$ 的误差。\n",
    "\n",
    "最小二乘模型下，更新策略为：\n",
    "\n",
    "$$\n",
    "\\mathbf w^{(\\tau+1)} = \\mathbf w^{(\\tau)} + \\eta \\{t_n - \\mathbf w^{(\\tau)\\top} \\phi(\\mathbf x_n)\\}\\phi(\\mathbf x_n)\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3.1.4 带正则的最小二乘"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "在 [1.1](../Chap-01-Introduction/01-01-Example-Polynomial-Curve-Fitting.ipynb) 节，我们引入了正则项来防止过拟合的发生，此时总的损失函数为：\n",
    "\n",
    "$$\n",
    "E_D(\\mathbf w) + \\lambda E_W(\\mathbf w)\n",
    "$$\n",
    "\n",
    "其中 $\\lambda$ 是正则项的系数，控制着 $E_D(\\mathbf w)$ 和 $E_W(\\mathbf w)$ 的相对重要程度。对于正则项，一个常用的选择是参数的平方和：\n",
    "\n",
    "$$\n",
    "E_W(\\mathbf w) = \\frac 1 2 \\mathbf{w^\\top w}\n",
    "$$\n",
    "\n",
    "此时，如果我们使用平方和误差：\n",
    "\n",
    "$$\n",
    "E_D(\\mathbf w) = \\frac{1}{2}\\sum_{n=1}^N \\{t_n-\\mathbf w^\\top\\mathbf \\phi(\\mathbf x_n)\\}^2\n",
    "$$\n",
    "\n",
    "那么，总的误差函数为\n",
    "\n",
    "$$\n",
    "\\frac{1}{2}\\sum_{n=1}^N \\{t_n-\\mathbf w^\\top\\mathbf \\phi(\\mathbf x_n)\\}^2 + \\frac \\lambda 2 \\mathbf{w^\\top w}\n",
    "$$\n",
    "\n",
    "事实上，这个正则项在机器学习领域对应于权值衰减（`weight decay`）的概念。权值衰减对非零项的权值进行惩罚，使得权值尽可能地接近零。\n",
    "\n",
    "因为上面的式子仍然是一个关于 $\\mathbf w$ 的二次项，因此我们不难应用之前的结论得到解为\n",
    "\n",
    "$$\n",
    "\\mathbf w=\\left(\\lambda \\mathbf I + \\mathbf{\\Phi^\\top \\Phi}\\right)^{-1} \\mathbf\\Phi^\\top\\mathsf t\n",
    "$$\n",
    "\n",
    "更一般的，我们可以使用这样的正则项形式：\n",
    "\n",
    "$$\n",
    "\\frac{1}{2}\\sum_{n=1}^N \\{t_n-\\mathbf w^\\top\\mathbf \\phi(\\mathbf x_n)\\}^2 + \\frac \\lambda 2 \\sum_{j=1}^M |w_j|^q\n",
    "$$\n",
    "\n",
    "$q=2$ 对应于二次正则项的形式。不同 $q$ 值对应的等高曲线如下。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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nmh8vcwL3ePBK7C/X6GFv/eXBXwH+mR/vkxc7WIx0Ir8OcIhFexW4DLgQuNKD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\nEREREZFG+H96IaAqOuXFJAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fad9436e510>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "_, axes = plt.subplots(1, 4, figsize=(14, 3))\n",
    "\n",
    "qq = [0.5, 1, 2, 4]\n",
    "\n",
    "for q, ax in zip(qq, axes):\n",
    "    ax.axis('off')\n",
    "    xx = np.linspace(0, 1)\n",
    "    yy = (1 - xx ** q) ** (1.0 / q)\n",
    "    ax.plot(xx, yy, 'g',\n",
    "            xx, -yy, 'g',\n",
    "            -xx, yy, 'g',\n",
    "            -xx, -yy, 'g')\n",
    "    ax.plot([-1.2, 1.2], [0, 0], 'k')\n",
    "    ax.plot([0, 0], [-1.2, 1.2], 'k')\n",
    "    ax.set_title('$q={}$'.format(q), fontsize='xx-large')\n",
    "    \n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$q=1$ 的情况在统计中又叫做 `lasso`。\n",
    "\n",
    "当 $\\lambda$ 足够大的时候，某些系数 $w_j$ 被强制趋于 0，这使得我们的模型趋于稀疏。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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vzCJFxMVm04WqghmdrF/vTqL/ihXuNB//9FPZ+HoZFZsdd9wBo0fbvxc891x3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\nCKhgKoqiKEoIqGAqiqIoSgj8PyE19AAe+3uhAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fad926bff10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "_, axes = plt.subplots(1, 2, figsize=(8, 5))\n",
    "\n",
    "qq = [2, 1]\n",
    "rr = [np.sqrt(4.25)-1, np.sqrt(1.25)]\n",
    "for q, ax, r in zip(qq, axes, rr):\n",
    "    ax.axis('equal')\n",
    "    xx = np.linspace(0, 1)\n",
    "    yy = (1 - xx ** q) ** (1.0 / q)\n",
    "    ax.plot(xx, yy, 'r',\n",
    "            xx, -yy, 'r',\n",
    "            -xx, yy, 'r',\n",
    "            -xx, -yy, 'r')\n",
    "    \n",
    "    ax.fill_between(xx, yy, -yy, color='gold')\n",
    "    ax.fill_between(-xx, yy, -yy, color='gold')\n",
    "    \n",
    "    tt = np.linspace(0, 2 * np.pi, 100)\n",
    "    for t in np.linspace(0.01, r, 4):\n",
    "        ax.plot(0.5 + t * np.sin(tt), 2 + t * np.cos(tt), 'b')\n",
    "    \n",
    "    ax.spines['right'].set_color('none')\n",
    "    ax.spines['top'].set_color('none')\n",
    "    \n",
    "    ax.xaxis.set_ticks_position('bottom')\n",
    "    ax.spines['bottom'].set_position(('data',0))\n",
    "\n",
    "    ax.yaxis.set_ticks_position('left')\n",
    "    ax.spines['left'].set_position(('data',0))\n",
    "    \n",
    "    ax.set_xlim([-1.5, 2])\n",
    "    ax.set_ylim([-1.5, 4])\n",
    "    \n",
    "    ax.set_xticks([])\n",
    "    ax.set_yticks([])\n",
    "    \n",
    "    ax.set_title('$q={}$'.format(q), fontsize='xx-large')\n",
    "    \n",
    "axes[0].text(0.2, 1.05, r'$w^\\star$', fontsize='x-large')\n",
    "axes[1].text(0.1, 1, r'$w^\\star$', fontsize='x-large')\n",
    "\n",
    "axes[0].plot(0.22, 0.97, 'ko')\n",
    "axes[1].plot(0, 1, 'ko')\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3.1.5 多维输出"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "之前我们考虑的是一维目标的情况，现在假设目标是 $K(K>1)$ 维的，记为 $\\bf t$。\n",
    "\n",
    "最简单的想法，我们考虑对于 $\\bf t$ 的每个分量使用一组不同的基函数，即将这个问题拆分为 $K$ 个独立的回归问题。\n",
    "\n",
    "不过更通常的做法是，我们使用同一组基函数来对所有的分量进行建模：\n",
    "\n",
    "$$\n",
    "\\bf y(x,w)=W^\\top\\phi(x)\n",
    "$$\n",
    "\n",
    "其中 $\\bf y$ 是 $K$ 维向量，$\\bf W$ 是 $M\\times K$ 的参数矩阵，$\\bf \\phi(x)$ 是 $M$ 维基函数向量，每个元素为 $\\mathbf\\phi_j(\\mathbf x)$，且 $\\mathbf\\phi_0(\\mathbf x)=1$。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 概率建模"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "我们考虑使用独立同分布的高斯分布对目标向量的条件分布进行建模：\n",
    "\n",
    "$$\n",
    "p(\\mathbf{t|x,W},\\beta)={\\mathcal N}({\\bf t|W^\\top\\phi(x)},\\beta^{-1}\\mathbf I)\n",
    "$$\n",
    "\n",
    "假设我们有 $N$ 个观测值 $\\mathbf t_1, \\cdots, \\mathbf t_N$，将转置后按行合并为一个 $N \\times K$ 的矩阵 $\\bf T$，即矩阵的第 $n$ 行为 $\\mathbf t_n^\\top$。类似的，我们将输入 $\\mathbf x_1, \\cdots, \\mathbf x_N$ 合并为一个矩阵 $\\bf X$。\n",
    "\n",
    "对数似然为\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\ln p(\\mathbf T|\\mathbf X,\\mathbf W,\\beta) & \n",
    "= \\sum_{n=1}^N \\ln \\mathcal N({\\mathbf t_n|\\mathbf W^\\top\\mathbf \\phi(\\mathbf x_n)},\\beta^{-1}\\mathbf I) \\\\\n",
    "& = \\frac{NK}{2} \\ln\\left(\\frac{\\beta}{2\\pi}\\right) - \\frac{\\beta}{2} \\sum_{n=1}^N\\left\\|\\mathbf t_n - \\mathbf W^\\top \\mathbf\\phi(\\mathbf x_n)\\right\\| \n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "不难得出，$W$ 的最大似然解为：\n",
    "\n",
    "$$\n",
    "\\mathbf W_{ML}=(\\mathbf{\\Phi^\\top\\Phi})^{-1} \\mathbf{\\Phi^\\top T}\n",
    "$$\n",
    "\n",
    "对于单个分量 $t_k$，我们有：\n",
    "\n",
    "$$\n",
    "\\mathbf w_{k}=(\\mathbf{\\Phi^\\top\\Phi})^{-1} \\mathbf{\\Phi^\\top} \\mathsf t_k = \\mathbf\\Phi^\\dagger \\mathsf t_k \n",
    "$$\n",
    "\n",
    "与我们一维输出的结果一致。"
   ]
  }
 ],
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   "display_name": "Python 3",
   "language": "python",
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    "name": "ipython",
    "version": 3
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   "file_extension": ".py",
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   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
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